# Eigenfunction Matching for a Finite Change in Depth

## Introduction

The problem consists of a region of free water surface with depth $\displaystyle{ h }$ except between $\displaystyle{ -L }$ and $\displaystyle{ L }$ where the depth is $\displaystyle{ d }$. The problem with a semi-infinite change in depth is treated in Eigenfunction Matching for a Semi-Infinite Change in Depth

## Governing Equations

We begin with the Frequency Domain Problem for a submerged dock which occupies the region $\displaystyle{ x\gt 0 }$ (we assume $\displaystyle{ e^{-\mathrm{i}\omega t} }$ time dependence). The depth of is constant $\displaystyle{ h }$ for $\displaystyle{ x\lt 0 }$ and constant $\displaystyle{ d }$ for $\displaystyle{ x\gt 0 }$. The $\displaystyle{ z }$-direction points vertically upward with the water surface at $\displaystyle{ z=0 }$. The boundary value problem can therefore be expressed as

$\displaystyle{ \Delta\phi=0, \,\, -h\lt z\lt 0, \,\, x\lt -L,\,x\gt L }$
$\displaystyle{ \Delta\phi=0, \,\, -d\lt z\lt 0, \,\, -L\lt x\lt L }$
$\displaystyle{ \partial_z\phi=\alpha\phi, \,\, z=0, }$
$\displaystyle{ \partial_x\phi=0, \,\, -d\lt z\lt -h,\,x=\pm L, }$
$\displaystyle{ \partial_z\phi=0, \,\, z=-h,\, x\lt -L,\,x\gt L }$
$\displaystyle{ \partial_z\phi=0, \,\, z=-d,\, -L\lt x\lt L }$

We must also apply the Sommerfeld Radiation Condition as $\displaystyle{ |x|\rightarrow\infty }$. This essentially implies that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave and a wave propagating away.

## Solution Method

We use separation of variables in the two regions, $\displaystyle{ x\lt 0 }$ and $\displaystyle{ x\gt 0 }$.

We express the potential as

$\displaystyle{ \phi(x,z) = X(x)Z(z)\, }$

and then Laplace's equation becomes

$\displaystyle{ \frac{X^{\prime\prime}}{X} = - \frac{Z^{\prime\prime}}{Z} = k^2 }$

### Separation of variables for a free surface

We use separation of variables

We express the potential as

$\displaystyle{ \phi(x,z) = X(x)Z(z)\, }$

and then Laplace's equation becomes

$\displaystyle{ \frac{X^{\prime\prime}}{X} = - \frac{Z^{\prime\prime}}{Z} = k^2 }$

The separation of variables equation for deriving free surface eigenfunctions is as follows:

$\displaystyle{ Z^{\prime\prime} + k^2 Z =0. }$

subject to the boundary conditions

$\displaystyle{ Z^{\prime}(-h) = 0 }$

and

$\displaystyle{ Z^{\prime}(0) = \alpha Z(0) }$

We can then use the boundary condition at $\displaystyle{ z=-h \, }$ to write

$\displaystyle{ Z = \frac{\cos k(z+h)}{\cos kh} }$

where we have chosen the value of the coefficent so we have unit value at $\displaystyle{ z=0 }$. The boundary condition at the free surface ($\displaystyle{ z=0 \, }$) gives rise to:

$\displaystyle{ k\tan\left( kh\right) =-\alpha \, }$

which is the Dispersion Relation for a Free Surface

The above equation is a transcendental equation. If we solve for all roots in the complex plane we find that the first root is a pair of imaginary roots. We denote the imaginary solutions of this equation by $\displaystyle{ k_{0}=\pm ik \, }$ and the positive real solutions by $\displaystyle{ k_{m} \, }$, $\displaystyle{ m\geq1 }$. The $\displaystyle{ k \, }$ of the imaginary solution is the wavenumber. We put the imaginary roots back into the equation above and use the hyperbolic relations

$\displaystyle{ \cos ix = \cosh x, \quad \sin ix = i\sinh x, }$

to arrive at the dispersion relation

$\displaystyle{ \alpha = k\tanh kh. }$

We note that for a specified frequency $\displaystyle{ \omega \, }$ the equation determines the wavenumber $\displaystyle{ k \, }$.

Finally we define the function $\displaystyle{ Z(z) \, }$ as

$\displaystyle{ \chi_{m}\left( z\right) =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0 }$

as the vertical eigenfunction of the potential in the open water region. From Sturm-Liouville theory the vertical eigenfunctions are orthogonal. They can be normalised to be orthonormal, but this has no advantages for a numerical implementation. It can be shown that

$\displaystyle{ \int\nolimits_{-h}^{0}\chi_{m}(z)\chi_{n}(z) \mathrm{d} z=A_{n}\delta_{mn} }$

where

$\displaystyle{ A_{n}=\frac{1}{2}\left( \frac{\cos k_{n}h\sin k_{n}h+k_{n}h}{k_{n}\cos ^{2}k_{n}h}\right). }$

### Inner product between free surface and dock modes

$\displaystyle{ B_{mn} = \int\nolimits_{-h}^{0}\phi_{m}^{d}(z)\phi_{n}^{h}(z) \mathrm{d} z }$

where

$\displaystyle{ B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(k_{m}^{h}(z+h))\cos(k_{n}^{d}(z+d))}{\cos(k_{m}h)\cos(k_{n}d)} \mathrm{d} z =\frac{k_{m}^{h}\sin(k_{m}^{d} h)\cos(k_{n}^{d}h)-k_{n}^{d}\cos(k_{m}^{h} h)\sin(k_{n}^{d}h)} {\cos(k_{m}h)\cos(k_{n}d)({k_{m}^{d}}^{2}-{k_{n}^{d}}^{2})} }$

## Governing Equations

We begin with the Frequency Domain Problem for a submerged dock which occupies the region $\displaystyle{ x\gt 0 }$ (we assume $\displaystyle{ e^{i\omega t} }$ time dependence). The depth of is constant $\displaystyle{ h }$ for $\displaystyle{ x\lt -L }$ and $\displaystyle{ x\gt L }$ and constant $\displaystyle{ d }$ for $\displaystyle{ -L\lt x\gt L }$. The $\displaystyle{ z }$-direction points vertically upward with the water surface at $\displaystyle{ z=0 }$. The boundary value problem can therefore be expressed as

$\displaystyle{ \Delta\phi=0, \,\, -h\lt z\lt 0, \,\, x\lt -L\,\textrm{or}\, x\gt L, }$
$\displaystyle{ \Delta\phi=0, \,\, -d\lt z\lt 0, \,\, -L\lt x\lt L, }$
$\displaystyle{ \partial_z\phi=\alpha\phi, \,\, z=0, }$
$\displaystyle{ \partial_x\phi=0, \,\, -d\lt z\lt -h,\,x=\pm{L}, }$
$\displaystyle{ \partial_z\phi=0, \,\, z=-h,\, x\lt -L\,\textrm{or}\, x\gt L, }$
$\displaystyle{ \partial_z\phi=0, \,\, z=-d,\, -L\lt x\lt L. }$

We must also apply the Sommerfeld Radiation Condition as $\displaystyle{ |x|\rightarrow\infty }$. This essentially implies that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave and a wave propagating away.

## Solution Method

We use separation of variables in the two regions, $\displaystyle{ x\lt 0 }$ and $\displaystyle{ x\gt 0 }$.

We express the potential as

$\displaystyle{ \phi(x,z) = X(x)Z(z)\, }$

and then Laplace's equation becomes

$\displaystyle{ \frac{X^{\prime\prime}}{X} = - \frac{Z^{\prime\prime}}{Z} = k^2 }$

### Separation of variables for a free surface

We use separation of variables

We express the potential as

$\displaystyle{ \phi(x,z) = X(x)Z(z)\, }$

and then Laplace's equation becomes

$\displaystyle{ \frac{X^{\prime\prime}}{X} = - \frac{Z^{\prime\prime}}{Z} = k^2 }$

The separation of variables equation for deriving free surface eigenfunctions is as follows:

$\displaystyle{ Z^{\prime\prime} + k^2 Z =0. }$

subject to the boundary conditions

$\displaystyle{ Z^{\prime}(-h) = 0 }$

and

$\displaystyle{ Z^{\prime}(0) = \alpha Z(0) }$

We can then use the boundary condition at $\displaystyle{ z=-h \, }$ to write

$\displaystyle{ Z = \frac{\cos k(z+h)}{\cos kh} }$

where we have chosen the value of the coefficent so we have unit value at $\displaystyle{ z=0 }$. The boundary condition at the free surface ($\displaystyle{ z=0 \, }$) gives rise to:

$\displaystyle{ k\tan\left( kh\right) =-\alpha \, }$

which is the Dispersion Relation for a Free Surface

The above equation is a transcendental equation. If we solve for all roots in the complex plane we find that the first root is a pair of imaginary roots. We denote the imaginary solutions of this equation by $\displaystyle{ k_{0}=\pm ik \, }$ and the positive real solutions by $\displaystyle{ k_{m} \, }$, $\displaystyle{ m\geq1 }$. The $\displaystyle{ k \, }$ of the imaginary solution is the wavenumber. We put the imaginary roots back into the equation above and use the hyperbolic relations

$\displaystyle{ \cos ix = \cosh x, \quad \sin ix = i\sinh x, }$

to arrive at the dispersion relation

$\displaystyle{ \alpha = k\tanh kh. }$

We note that for a specified frequency $\displaystyle{ \omega \, }$ the equation determines the wavenumber $\displaystyle{ k \, }$.

Finally we define the function $\displaystyle{ Z(z) \, }$ as

$\displaystyle{ \chi_{m}\left( z\right) =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0 }$

as the vertical eigenfunction of the potential in the open water region. From Sturm-Liouville theory the vertical eigenfunctions are orthogonal. They can be normalised to be orthonormal, but this has no advantages for a numerical implementation. It can be shown that

$\displaystyle{ \int\nolimits_{-h}^{0}\chi_{m}(z)\chi_{n}(z) \mathrm{d} z=A_{n}\delta_{mn} }$

where

$\displaystyle{ A_{n}=\frac{1}{2}\left( \frac{\cos k_{n}h\sin k_{n}h+k_{n}h}{k_{n}\cos ^{2}k_{n}h}\right). }$

## Solution using Symmetry

The finite dock problem is symmetric about the line $\displaystyle{ x=0 }$ and this allows us to solve the problem using symmetry. This method is numerically more efficient and requires only slight modification of the code for Eigenfunction Matching for a Semi-Infinite Dock, the developed theory here is very close to the semi-infinite solution. We decompose the solution into a symmetric and an anti-symmetric part as is described in Symmetry in Two Dimensions

### Symmetric solution

The symmetric potential can be expanded as

$\displaystyle{ \phi(x,z)=e^{-k_{0}^{h}(x+L)}\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m}^{s}e^{k_{m}^{h}(x+L)}\phi_{m}^{h}(z) , \;\;x\lt -L }$

and

$\displaystyle{ \phi(x,z)=\sum_{m=0}^{\infty}b_{m}^{s} \frac{\cosh k_{m}^{d} x}{\cosh k_{m}^{d} L} \phi_{m}^{d}(z), \;\;-L\lt x\lt 0 }$

where $\displaystyle{ a_{m}^{s} }$ and $\displaystyle{ b_{m}^{s} }$ are the coefficients of the potential in the two regions.

For the first equation we multiply both sides by $\displaystyle{ \phi_{n}^{h}(z) \, }$ and integrate from $\displaystyle{ -h }$ and for the second equation we multiply both sides by $\displaystyle{ \phi_{n}^{d}(z) \, }$ and integrate from $\displaystyle{ -d }$. This gives us

$\displaystyle{ A_{0}\delta_{0n}+a_{n}^{s}A_{n}^{h} =\sum_{m=0}^{\infty}b_{m}^{s}B_{mn} }$

and

$\displaystyle{ -k_{0}^{h}B_{n0} + \sum_{m=0}^{\infty} a_{m}^{s}k_{m}^{h} B_{nm} = -b_{n}^{s}k_{n}^{d}\tanh(k_{n}^{d}L) A_{n}^{d} }$

(for full details of this derivation see Eigenfunction Matching for a Semi-Infinite Change in Depth)

### Anti-symmetric solution

The anti-symmetric potential can be expanded as

$\displaystyle{ \phi(x,z)=e^{-k_{m}^{h}(x+L)}\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m}^{a}e^{k_{m}^{h}(x+L)}\phi_{m}(z) , \;\;x\lt -L }$

and

$\displaystyle{ \phi(x,z)=\sum_{m=0}^{\infty}b_{m}^{a} \frac{\sinh k_{m}^{h} x}{-\sinh k_{m}^{h} L}\phi_{m}(z), \;\;-L\lt x\lt 0 }$

where $\displaystyle{ a_{m}^{a} }$ and $\displaystyle{ b_{m}^{a} }$ are the coefficients of the potential in the two regions. Note that the minus sign in the expression for the dock-covered region has been added so that each component is equal to one at $\displaystyle{ x=-L }$.

For the first equation we multiply both sides by $\displaystyle{ \phi_{n}^{h}(z) \, }$ and integrate from $\displaystyle{ -h }$ and for the second equation we multiply both sides by $\displaystyle{ \phi_{n}^{d}(z) \, }$ and integrate from $\displaystyle{ -d }$. This gives us

$\displaystyle{ A_{0}^{h}\delta_{0n}+a_{n}^{a}A_{n}^{h} =\sum_{m=0}^{\infty}b_{m}^{a}B_{mn} }$

and

$\displaystyle{ -k_{0}^{h}B_{n0} + \sum_{m=0}^{\infty} a_{m}^{a}k_{m}^{h} B_{nm} = -b_{n}^{a}k_{n}^{d}\coth(k_{n}^{d}L) A_{n}^{d} }$

(for full details of this derivation see Eigenfunction Matching for a Semi-Infinite Change in Depth)

### Solution to the original problem

We can now reconstruct the potential for the finite dock from the two previous symmetric and anti-symmetric solution as explained in Symmetry in Two Dimensions. The amplitude in the left open-water region is simply obtained by the superposition principle

$\displaystyle{ a_{m} = \frac{1}{2}\left(a_{m}^{s}+a_{m}^{a}\right) }$

$\displaystyle{ d_{m} = \frac{1}{2}\left(a_{m}^{s}-a_{m}^{a}\right) }$

Note the formulae for $\displaystyle{ b_m }$ and $\displaystyle{ c_m }$ are more complicated but can be derived with some work.

## Matlab Code

A program to calculate the coefficients for the semi-infinite dock problems can be found here finite_change_in_depth.m